Question

第一幅图中的正方形绕固定的正六边形顺时针“滚动”，直到到达底部。在图 $4$ 中，实心三角形会处在哪个位置？

The square in the first diagram “rolls” clockwise around the fixed regular hexagon until it reaches the bottom. In which position will the solid triangle be in diagram $4?$

答案：A

知识点：变换正多边形

难度评级：1510

小提示：

正方形每沿六边形的一条边滚动一次，都会绕共同顶点顺时针转 $150^\circ$ 。

Each time the square rolls over one edge of the hexagon it pivots about a shared corner and turns $150^\circ$ clockwise

大提示：

从顶边到底边需要滚动 $3$ 次，总共转 $3 \times 150^\circ = 450^\circ$ ，也就是转一整圈后再顺时针多转四分之一圈

Going from the top edge to the bottom edge is $3$ rolls, a total of $3 \times 150^\circ = 450^\circ,$ which is one full turn plus an extra quarter turn clockwise

解答：

正方形每滚过六边形的一条边，就顺时针转 $150^\circ$ 。（在旋转顶点处，正方形转过 $360^\circ$ 减去自身 $90^\circ$ 的角，再减去六边形 $120^\circ$ 的内角，剩下 $150^\circ$ 。）

从六边形顶部到底部需要滚动 $3$ 次，总转角为 $3 \times 150^\circ = 450^\circ$ 。这是一整圈（ $360^\circ$ ）再加顺时针 $90^\circ$ 。

三角形开始时指向正上方，所以额外顺时针转四分之一圈后，它指向右方。

所以正确答案是 A。

Each roll of the square over one edge of the hexagon turns it $150^\circ$ clockwise. (At the pivot corner the square turns through $360^\circ$ minus the square’s own $90^\circ$ corner minus the hexagon’s $120^\circ$ interior angle, which leaves $150^\circ.$ )

From the top of the hexagon to the bottom is $3$ rolls, for a total of $3 \times 150^\circ = 450^\circ.$ That is one complete revolution ( $360^\circ$ ) plus an extra $90^\circ$ clockwise.

The triangle starts pointing straight up, so after the extra quarter turn clockwise it points to the right.

Thus, the correct answer is A .

1988 AMC 8 第 24 题

答案：A

难度评级：1510

小提示：

大提示：

解答：

其他年份的第 24 题